We introduce and study a family of axioms that closely follows the pattern of parametrized diamonds, studied by Moore, Hru\v{ s}\'ak, and D\v{ z}amonja in [13]. However, our approach appeals to model theoretic / forcing theoretic notions, rather than pure combinatorics. The main goal of the paper is to exhibit a surprising, close connection between seemingly very distinct principles. As an application, we show that forcing with a measure algebra preserves (a variant of) $\diamondsuit(\mathfrak{ d})$, improving an old result of M. Hru\v{ s}\'ak.
Categories: math.LO